isolation theorem for products of linear forms - American Mathematical

PROCEEDINGS OF THE AMERICAN MATHEMATICAL Volume

100, Number

SOCIETY

1, May 1987

ISOLATION THEOREM FOR PRODUCTS

OF LINEAR FORMS

T. W. CUSICK ABSTRACT. A theorem of Cassels and Swinnerton-Dyer about products of three linear forms with real coefficients is generalized to products of any number of linear forms.

1. Introduction. In this paper we generalize some results of Cassels and Swinnerton-Dyer [3]. Suppose /(xi,...,xn) is a product of n > 3 linear forms with real coefficients. Our first theorem is an isolation theorem for these forms f{x\,... ,xn). For a discussion of the significance of isolation theorems in the geometry of numbers see Cassels [2, pp. 264-265]; for examples and applications of

isolation theorems see [2, pp. 286-298]. In order /. For any such that coefficients

to state our theorem we need to define an ¿-neighborhood of the form e > 0, an e-neighborhood of / is the set of all products of n linear forms the coefficients of the linear forms are within e of the corresponding of the linear forms in /. Any set which contains some e-neighborhood

of / will be called a neighborhood of /. THEOREM l.

Let f{xi,...,

xn) be the product of n > 3 linear forms with real

coefficients. Suppose that f has integer coefficients and that / = 0 only when all the Xi are 0. Let {61,62) be any open interval. Then there is a neighborhood of f such that all forms in the neighborhood which are not multiples of f itself take some value in the interval {61,62) for some integer values of the variables Xi,... ,xn.

It is well known (see [2, pp. 285-286]) that the conditions imposed on the product / of linear forms in Theorem 1 imply that / is equal to an integer times the product of all the n conjugates of one linear form whose coefficients are algebraic integers in some totally real algebraic number field of degree n. Thus Theorem 1 is really a statement about norm forms. Of course any norm form / satisfies inf |/| > 0, where the infimum is taken over all integers xi,... ,x„ not all zero. It is a notorious unsolved problem [2, pp. 260-264] to decide whether norm forms are the only products of n > 3 linear forms with this property. For n — 2, this problem has a negative answer because we can find a binary quadratic form

f{x, y) = (x + 0iy){x + 62y) = x2 + ßxy + iy2 with 7 irrational such that inf \f{x, y)\ > 1 for x,y not both zero. Such forms exist as long as /32 — 47 > 9, and indeed there are uncountably many with ß2 —4^y = 9 (see Cassels [1, Lemma 14, pp. 38-39]). Also, the analog of Theorem 1 is false for n = 2, for instance if we consider the form x2 — 2>y2. A certain weaker isolation

theorem [2, pp. 287-289] is valid for n = 2. Received by the editors September 26, 1985 and, in revised form, March 25, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 10E15; Secondary 10C10, 10F99. Key words and phrases. Linear

forms, norm forms. ©1987

American

0002-9939/87

29

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T. W. CUSICK

30

The case n — 3 of Theorem 1 was proved by Cassels and Swinnerton-Dyer [3, Theorem 2, p. 74]. Skubenko [4, 5] also claims a proof of Theorem 1, but his argument is difficult to follow (in particular, when he uses Lemma 3, Corollary in [4] to prove the theorem of that paper, he needs to show, in his notation, that jj2£i/2 __>Q ^ g __,.q^ hU£ £njs js no^. proved). In any case, the proof of Theorem 1 given here is simpler than that of Skubenko; indeed, our proof is along the lines of the original proof of the case n — 3 [3, pp. 78-79], but with one additional idea.

2. Proof

of Theorem

1. We first require a generalization

of Lemma 1 of [3].

LEMMA 1. Fix m > 2. Let A = [oíj] be an m by m matrix of real numbers with det A ^ 0. Suppose not all of the numbers a\ja^ {j = 2,3,... ,m) are rational. Then given any r > 0 there exists a a, depending on r and on the üíj, such that for any A there are integers «i,..., um such that \uia\i

+ u2a\2 + ■■■+ umaim

- A| < r

and \uian

+ u2a,i2 H-h

umaim \ < a

for i = 2,3,...,

m.

PROOF. This is an easy consequence of Kronecker's theorem on Diophantine approximation. In order to apply Lemma 1 in the proof of Theorem 1, we shall need the following lemma about units in totally real fields. If a is a number in an algebraic number field of degree d, we let a = a^\a^2\ ..., a^ denote the conjugates of a. The elegant proof of Lemma 2, due to Swinnerton-Dyer, replaces a clumsier one of the author. LEMMA 2. Suppose K is a totally real algebraic number field of degree r > 3 and Xii ■■■iXr-i are any multiplicatively independent units in K. Then for any

i 5¿j, 1 < i,j < r, such that \x[ | f^ IXi l> the r - 2 numbers

log|x^/x£j)|

(fc_23

r_t)

are not all rational. PROOF. We suppose that for some fixed i, j and rational

integers p{k), q{k) ^ 0

we have

(1)

loglx^l

P{k)

3

we shall deduce a contradiction. We start with some simplifications, none of which alter the multiplicative independence of the Xk- First we square each Xk\ then (1) still holds and we can remove the absolute value signs in (1) since every Xfcis totally positive. Now we define units

(2)

yk = xl{k)x7ik)

since q{k) ^ 0, the yk {k = 2,3,...,

(fc= 2,3,...,r-l); r — 1) are multiplicatively

independent.

(1) gives

lorflíW) whence t/[

= Q(k)log(xilVxiJ))- P(fc)log(x^/XiJ)) = 0,

= yk3 since yk is totally positive.

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Now

ISOLATION THEOREM

FOR PRODUCTS

OF LINEAR FORMS

31

Now let L be the subfield of K made up of those b such that b^ =bfj>; since L is a proper subfield of K, it has degree at most kr. But (2) gives r —2 multiplicatively independent units of L, so we obtain

r-2